Linear difference inequalities with constant coefficients with the sum equal to zero

Abstract

Many difference equations of the form x<inf>n</inf>+<inf>k</inf> = f(x<inf>n</inf>+<inf>k</inf><inf>−1</inf>, . . ., x<inf>n</inf>), n ∈ N, where k ∈ N, model some phenomena in nature and society. The most interesting cases usually occur when the function f satisfies the condition f(x, . . ., x) = x on its domain of definition. Because of this the difference equations x<inf>n</inf>+<inf>k</inf> ≤ f(x<inf>n</inf>+<inf>k</inf><inf>−1</inf>, . . ., x<inf>n</inf>) and x<inf>n</inf>+<inf>k</inf> ≥ f(x<inf>n</inf>+<inf>k</inf><inf>−1</inf>, . . ., x<inf>n</inf>) are of some interest. If f is a smooth function, then it can be approximated by a linear function. Motivated by some concrete examples, here we mostly consider the sequences that satisfy the linear difference inequality k X a<inf>j</inf>x<inf>n</inf>+<inf>l</inf><inf>− j</inf> ≥ 0, n ∈ N<inf>0</inf>, j=1 where k ∈ N<inf>2</inf>, l ∈ N<inf>0</inf>, and the coefficients a<inf>j</inf> ∈ R, j = 2, k − 1, a<inf>1</inf>, a<inf>k</inf> ∈ R \ {0}, satisfy the condition ^Pk<inf>j</inf><inf>=1</inf> a<inf>j</inf> = 0.